Outer Functions and Divergence in de Branges–Rovnyak Spaces

Abstract

In most classical holomorphic function spaces on the unit disk in which the polynomials are dense, a function f can be approximated in norm by its dilates $f_r(z):=f(rz)~(r<1)$, in other words, $\lim _{r\rightarrow 1^-}\Vert f_r-f\Vert =0$. We construct a de Branges–Rovnyak space $\mathcal{H}(b)$ in which the polynomials are dense, and a function $f\in \mathcal{H}(b)$ such that $\lim _{r\rightarrow 1^-}\Vert f_r\Vert _{\mathcal{H}(b)}=\infty$. The essential feature of our construction lies in the fact that b is an outer function.